FM for Actuaries

Loans and Costs of Borrowing 165
the monthly-rest loan, which shows that the monthly-rest loan is preferred. On the
other hand, the ENR of the 20-year annual-rest loan is lower than the quoted rate of
the monthly-rest loan, which means that the annual-rest loan should be preferred.
This is consistent with the answer in Example 5.13.
It should be noted that the ENR r ∗ defined in (5.12) is actually r (12) defined
in Chapter 1. If the installment is made m times a year, we can modify (5.11) to
define the ENR accordingly, and we shall denote it simply by r (m) . We conclude
this section by stating that the ENR is a nominal rate; it is not an effective rate as
defined in Chapter 1.
5.6 Flat Rate Loan and Flat Rate Discount Loan
Some short-term loans are charged on a flat rate basis. This means that interest is
charged on the original principal for the whole term of the loan, even though the
loan is progressively repaid by periodic installments. For a n-year loan of amount
L charged on a quoted flat rate of interest of r per annum, the monthly installment
A is computed by
L(1 + rn)
A = . (5.13)
12n
If the installments are paid in arrears, the loan is called a flat rate loan. To compute
the ENR of this loan, we solve for r (12) from the equation
or, equivalently,
Aa 12n⌉ r (12)
12
a 12n⌉ r (12)
12
= L,
= 12n
1+rn . (5.14)
Example 5.15: A bank charges a car loan at 3% per annum flat. If the loan period
is 5 years, and the repayments are by monthly installments in arrears, what is the
ENR? What is the effective rate of interest?
Solution: Using (5.14), we solve for r (12) in
a 60⌉ r (12)
12
=
12 × 5
1+0.03 × 5 =52.1739
to obtain r (12) =5.64%. The effective rate of interest is
(
1+ 0.0564 ) 12
− 1=5.79%.
12